Week 10: Maxwell’s Equations and Light 367
Problem 6.Let’s work out an interesting fact about the solar wind. Consider a spherical grain
of dust of radiusRwith a “reasonable” mass density of 1000 kg per cubic meter (the
density of water). Given the mass of the sun (see problem above),your knowledge ofG
(the gravitational constant) and the insight that the radiation pressure from sunlight is
approximately exerted on the transverse cross-sectional areaof the sphereπR^2 , determine
the radiusRcfor which the force exerted by light pressureawayfrom the sunexactly
balancesthe gravitational forcetowardsthe sun.
Will particles larger than this (smaller than this) fall into or be pushedaway from the
sun? Note well that this differential force is exerted no matter howfar away from the
sun one travels, so particles pushed away are accelarated all the way! This explains why
small particles (gas molecules, dust particles) are acceleratedawayfrom stars, forming
a constant “wind” of microparticle radiation.Problem 7.Suppose you have a long solenoid (of lengthL, withn=N/Lturns per unit length and
radiusR) carrying a time varying currentI(t) =I 0 (1−e−t/τ).a) FindBz(t) inside the solenoid.
b) Find the induced electrical field at an arbitrary point inside the solenoid (say, at a
distancerfrom its axis).
c) Find the magnitude and direction of the Poynting vector on an imagined surface of
constant radius just inside the windings at radiusR.
d) Compute the flux of the Poynting vector into the volume of the solenoid.
e) Compute the total magnetic energy of the solenoid, and show that the flux of the
Poynting vector equals the rate at which this energy changes.Problem 8.A vertical cell phone radio tower acts as a dipole antenna. Supposesuch a tower is
located 1 km away from your cell phone. It radiates a power of 1 kilowatt. What is
the approximate intensity of this radation when it reaches your phone? Now consider
your phone. It’s dipole antenna radiates roughly one watt when it operates. What is the
radiation intensity of your cell phone back at the tower?Problem 9.A capacitor consisting of twocircularconducting disks of radiusRis being charged by
a steady currentI. Find the magnetic and electric fields at an arbitrary point inside
the volume ofempty spacebetween the two plates (using Gauss’s Law and Ampere’s
Law with the Maxwell Displacement Current, respectively). Form the Poynting vector
at a point on the “boundary” of the E field, assuming no fringing fields, and integrates
the flux of the Poynting vector into the volume of the capacitor. Show that the result
equalsPC=VCI, the power being delivered to the capacitor. (Note this problem, the